关于无振荡、无自由参数有限元格式的研究

利用双曲守恒律方程的Ｔａｙｌｏｒ弱解表达式，建立了有限元法修正方程，选择合适的展开式系数能得到一系列数值格式．通过稳定性分析研究了格式的稳定性、色散误差与有限元修正方程导数项系数之间的关系，该关系与差分法的ＮＮＤ格式一致．在选定格式下，通过ＣＦＬ数可控制有限元离散解的振荡而使格式不含自由参数．最后，用数值算例验证了这一关系，并在二、三维欧拉方程作了推广应用．

RESEARCH ON NON OSCILLATION, PARAMETER FREE FINITE ELEMENT SCHEME 1)

A finite element modified equation is estabished by using the finite element Taylor weak statement for hyperbolic conservation laws and various finite element numerical schemes are obtained by means of reasonably choosing the coefficient of the expansional expression. The relations between the coefficients of the differential terms of the finite element modified equation and stability, dissipation error and dispersion error of finite element schemes are investigated by means of stability analysis. These relations are in conformity with NND schemes developed by reference for finite difference method. For a selected finite element scheme, the spurious oscillation of solution may be controlled by regulating the CFL number and make schemes free of parameter. Finally, linear and nonlinear numerical examples are carried out for verifications and the conclusion is applied to the solution of two and three dimensional Euler equations.